The present paper considers a diffusive Nicholson's blowflies model with multiple delays under a Neumann boundary condition. Delay independent conditions are derived for the global attractivity of the trivial equilibrium and the positive equilibrium, respectively. Two open problems concerning the stability of positive equilibrium and the occurrence of Hopf bifurcation are proposed.
Since blowflies are important parasites of the sheep industry in some countries such as Australia, based on the experimental data of Nicholson [
It is impossible that the size of the adult blowflies population is independent of a spatial variable; therefore, Yang and So [
Meanwhile, one can consider a nonlinear equation with several delays because of variability of the generation time; for this purpose, Györi and Ladas [
Luo and Liu [
It is of interest to investigate both several temporal and spatial variations of the blowflies population using mathematical models. Hereby, in this paper, we consider the following system:
Though the global attractivity of the nonnegative equilibria of (
It is not difficult to see that if
The rest of the paper is organized as follows. We give some lemmas and definitions in Section
In this section, we will give some lemmas which can be proved by using the similar methods as those in Yang and So [
(i) The solution
(ii) If
Next, we will introduce the concept of lower-upper solution due to Redlinger [
A lower-upper solution pair for (
The following lemma is a special case of Redlinger [
Let
The following lemma gives us boundedness of the solution
(i) The solution
(ii) There exists a constant
Let
Solving the equation, we have
Taking
By Lemma
Note that
Therefore, the formula (
So we complete Lemma
Assume that
By Lemma
Define
By using the similar methods to prove Lemma
Because of
Therefore, from Definition
By Theorem 1 of Luo and Liu [
Hence, we complete the proof of Theorem
If
Let the function
There are now two possible cases to consider.
Let
From Lemma
Let
We prove that
By induction and direct computation, we have
Similarly, we have
Define
It follows from (
Therefore, from Definition
Note that
Define
Repeating the above procedure, we have the following relation:
By (
Our main results are also valid when
In this section, we will give some numerical simulations to verify our main results in Section
Different parameters will be used for simulations, and some data come from [
Parameters:
Figure
Parameters:
In Section
It is similar to Theorem 3 in Luo and Liu [
If
Figure
Parameters:
From Figure
Parameters:
From Figure
Parameters:
Under suitable conditions, the systems (
Now, we have not intensively studied these two problems. Because the nonmonotonicity of the nonlinear term in (
Project was supported by Hunan Provincial Natural Science Foundation of China (12jj4012) and Research Project of National University of Defense Technology (JC12-02-01).